In this lesson we used long division and practiced dividing by binomials. It was generally a review and a refresher of things we've already learned in the past. One important theorem we learned is the Remainder Theorem. It states "If a polynomial f(x) is divided by x-c then the remainder is f(x)." -Shelby N.
Just with what shelby said, we devided polynomials by binomials using long devision. and like she said, it was mostly a review. there really isn't much to say about this lesson because it was pretty easy and was a review from what we learned in the past years.
-Jesse Blank
Lesson 9-4 deals with using long division to divide polynomials. Although synthetic division is much faster and easier to use, it can only be used when you are dividing by a linear factor. Dividing polynomials with long division may take a while, but it is very similar to dividing numbers. The main thing to do is to see how many times the first part of the divisor can go into the dividend, then multiply that answer with the second part of the divisor, subtract like terms, and repeat the process until you get a remainder. If you want to find the remainder of a division problem without dividing, you can use the remainder theorem, which is stated in the above entry.
-Nathan Felty
In this lesson we utilized our new knowledge of long division of polynomials to find factors. We learned a very important theorem, the The Factor Theorem: For a polynomial f(x), a number c is a solution to f(x)=0 IFF (x-c) is a factor of f.
We learned that to find zeros of a polynomial, you can use your calculator to find the first zero, then use long division to find the rest. This involves the steps to factoring polynomials that are:
1. a x c
2. factor ac -> a+c=b
3. replace b
4. group 1st two and last 2 terms
5. GCF
6. factors: stuff inside and stuff outside
Also, the factor are the inverse of what they are on the graph.
-Andrea M.
Factor Theorem!!-For a polynomial f(x), a number c is a solution to f(x) = 0 if and only if (x-c) is a factor of f.
Factor-Solution-Intercept Equivalence Theorem- For any polynomial f, the following are logically equivalent statements.
1. (x-c) is a factor of f.
2. f(c) = 0
3. c is an x-intercept of the graph of y = f(x).
4. c is a zero of f.
5. The remainder when f(x) is divided by (x-c).
-Jesse Blank
Synthetic Division is an easier way to divide linear polynomials and it doesn't use variables. It can also test if a possible root is an actual zero. To find all of the possible roots you use the Rational Roots Test. This is done by letting p be all of the factors of the leading coefficient and q be all of the factors of the constant in any polynomial. Then you simply divide p by q for all of the values. -Jeremy G
Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form x-a. It can be used in place of the standard long division algorithm. This method reduces the polynomial and the linear factor into a set of numeric values. After these values are found, the resulting set of numeric outputs is used to construct the polynomial quotient and the polynomial remainder. Terms you may need to know: dividend- a quanity that is divided by another quanity and diviso- the number or expression b when a is divided by b. -Katie K.___
In this lesson we learned that a similar theorem for the factorization of sums and differences of even powers does not exist. If n is a positive even integer, then x^n+y^n does not have a linear factor with real coefficients. To factor the difference of two powers, x^n-y^n for n even, think of the even power as the square of some lover power, and reduce the problem to the difference of the two squares.
example: x^6-64
x^6=(x^3)^2, and 64=2^6=(2^3)^2
=(x^3-2^3)(x^3+2^3)
-Kelly H.
Secant (sec), cosecant (csc), and cotangent (cot) can be expressed as the reciprocal of a parent trigonometric function, thus they are sometimes called reciprocal trigonometric functions. In order to solve for these functions you need to take the reciprocal. Such as
secθ = 1/cosθ, for cosθ≠0 The same is true for cscθ = 1/sinθ and cotθ = cosθ/sinθ, when sinθ≠0. Remember the reciprocal of these functions is NOT the inverse! - Katie K.
Table of Contents
Chapter Everything Else
9-4: Division and the Remainder Theorem
In this lesson we used long division and practiced dividing by binomials. It was generally a review and a refresher of things we've already learned in the past. One important theorem we learned is the Remainder Theorem. It states "If a polynomial f(x) is divided by x-c then the remainder is f(x)." -Shelby N.
Just with what shelby said, we devided polynomials by binomials using long devision. and like she said, it was mostly a review. there really isn't much to say about this lesson because it was pretty easy and was a review from what we learned in the past years.
-Jesse Blank
Lesson 9-4 deals with using long division to divide polynomials. Although synthetic division is much faster and easier to use, it can only be used when you are dividing by a linear factor. Dividing polynomials with long division may take a while, but it is very similar to dividing numbers. The main thing to do is to see how many times the first part of the divisor can go into the dividend, then multiply that answer with the second part of the divisor, subtract like terms, and repeat the process until you get a remainder. If you want to find the remainder of a division problem without dividing, you can use the remainder theorem, which is stated in the above entry.
-Nathan Felty
9-5: The Factor Theorem
In this lesson we utilized our new knowledge of long division of polynomials to find factors. We learned a very important theorem, the The Factor Theorem: For a polynomial f(x), a number c is a solution to f(x)=0 IFF (x-c) is a factor of f.
We learned that to find zeros of a polynomial, you can use your calculator to find the first zero, then use long division to find the rest. This involves the steps to factoring polynomials that are:
1. a x c
2. factor ac -> a+c=b
3. replace b
4. group 1st two and last 2 terms
5. GCF
6. factors: stuff inside and stuff outside
Also, the factor are the inverse of what they are on the graph.
-Andrea M.
Factor Theorem!!-For a polynomial f(x), a number c is a solution to f(x) = 0 if and only if (x-c) is a factor of f.
Factor-Solution-Intercept Equivalence Theorem- For any polynomial f, the following are logically equivalent statements.
1. (x-c) is a factor of f.
2. f(c) = 0
3. c is an x-intercept of the graph of y = f(x).
4. c is a zero of f.
5. The remainder when f(x) is divided by (x-c).
-Jesse Blank
Synthetic Division
Synthetic Division is an easier way to divide linear polynomials and it doesn't use variables. It can also test if a possible root is an actual zero. To find all of the possible roots you use the Rational Roots Test. This is done by letting p be all of the factors of the leading coefficient and q be all of the factors of the constant in any polynomial. Then you simply divide p by q for all of the values. -Jeremy G
Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form x-a. It can be used in place of the standard long division algorithm. This method reduces the polynomial and the linear factor into a set of numeric values. After these values are found, the resulting set of numeric outputs is used to construct the polynomial quotient and the polynomial remainder. Terms you may need to know: dividend- a quanity that is divided by another quanity and diviso- the number or expression b when a is divided by b. -Katie K.___
9-8: Factoring Sums and Differences of Powers
In this lesson we learned that a similar theorem for the factorization of sums and differences of even powers does not exist. If n is a positive even integer, then x^n+y^n does not have a linear factor with real coefficients. To factor the difference of two powers, x^n-y^n for n even, think of the even power as the square of some lover power, and reduce the problem to the difference of the two squares.
example: x^6-64
x^6=(x^3)^2, and 64=2^6=(2^3)^2
=(x^3-2^3)(x^3+2^3)
-Kelly H.
9-9: Advanced Factoring Techniques
13-1: The Secant, Cosecant, and Cotangent Functions
Secant (sec), cosecant (csc), and cotangent (cot) can be expressed as the reciprocal of a parent trigonometric function, thus they are sometimes called reciprocal trigonometric functions. In order to solve for these functions you need to take the reciprocal. Such as
secθ = 1/cosθ, for cosθ≠0 The same is true for cscθ = 1/sinθ and cotθ = cosθ/sinθ, when sinθ≠0. Remember the reciprocal of these functions is NOT the inverse! - Katie K.
Parametric Equations
Calendar